Direct Simulation Of Turbulent Flow Research Articles

Turbulent kinetic energy production and flow structures in flows past smooth and rough walls

Data available in the literature from direct numerical simulations of two-dimensional turbulent channels by Lee & Moser (J. Fluid Mech., vol. 774, 2015, pp. 395–415), Bernardini et al. (J. Fluid Mech., 742, 2014, pp. 171–191), Yamamoto & Tsuji (Phys. Rev. Fluids, vol. 3, 2018, 012062) and Orlandi et al. (J. Fluid Mech., 770, 2015, pp. 424–441) in a large range of Reynolds number have been used to find that $S^{\ast }$ the ratio between the eddy turnover time ($q^{2}/\unicode[STIX]{x1D716}$, with $q^{2}$ being twice the turbulent kinetic energy and $\unicode[STIX]{x1D716}$ the isotropic rate of dissipation) and the time scale of the mean deformation ($1/S$), scales very well with the Reynolds number in the wall region. The good scaling is due to the eddy turnover time, although the turbulent kinetic energy and the rate of isotropic dissipation show a Reynolds dependence near the wall; $S^{\ast }$, as well as $-\langle Q\rangle =\langle s_{ij}s_{ji}\rangle -\langle \unicode[STIX]{x1D714}_{i}\unicode[STIX]{x1D714}_{i}/2\rangle$ are linked to the flow structures, and also the latter quantity presents a good scaling near the wall. It has been found that the maximum of turbulent kinetic energy production $P_{k}$ occurs in the layer with $-\langle Q\rangle \approx 0$, that is, where the unstable sheet-like structures roll-up to become rods. The decomposition of $P_{k}$ in the contribution of elongational and compressive strain demonstrates that the two contributions present a good scaling. However, the good scaling holds when the wall and the outer structures are separated. The same statistics have been evaluated by direct simulations of turbulent flows in the presence of different types of corrugations on both walls. The flow physics in the layer near the plane of the crests is strongly linked to the shape of the surface and it has been demonstrated that the $u_{2}$ (normal to the wall) fluctuations are responsible for the modification of the flow structures, for the increase of the resistance and of the turbulent kinetic energy production.

10.1007/s11470-008-2012-4

A parallel software package designed for the numerical simulation of three-dimensional viscous gas flows is presented. The numerical algorithm is based on kinetically consistent difference schemes used on locally refined grids. The software package has been tested in various super-and subsonic flow problems. It provides an opportunity for the direct simulation of turbulent flows. The efficiency of parallelization is analyzed depending on the problem size and the number of processors.

Software package for 3D viscous gas flow simulation on multiprocessor computer systems

A parallel software package designed for the numerical simulation of three-dimensional viscous gas flows is presented. The numerical algorithm is based on kinetically consistent difference schemes used on locally refined grids. The software package has been tested in various super-and subsonic flow problems. It provides an opportunity for the direct simulation of turbulent flows. The efficiency of parallelization is analyzed depending on the problem size and the number of processors.

Multilevel Parallelization Models in CFD

*† We present two multilevel parallel models based on MPI/MPI (MPI denoting Message Passing Interface) and MPI/OpenMP (OpenMP denoting Open Multi-Processing) for highorder CFD methods and compare their performances. These models are implemented within the spectral/hp element framework to take advantage of the hierarchical structures arising from deterministic and stochastic CFD computations. For MPI/MPI, we employ MPI process groups to decompose the computations into different levels. For MPI/OpenMP, we take a Single-Program-Multiple-Data (SPMD) style approach to OpenMP shared memory parallelism that significantly reduces the OpenMP synchronizations. These models demonstrate a good scalability with respect to the problem size and a good speedup for fixed problem sizes. With identical configurations, the MPI/MPI parallel model is observed to be generally more efficient. The advantage of these multilevel approaches lies in that they reduce the number of processes participating in each communication and the latency overhead, and thus enable the applications to scale to a large number of processors more efficiently. The models have been applied to the direct simulation of turbulent flows past a circular cylinder at Reynolds number Re=10,000.

A Generic Model for Calculation of Frictional Losses in Pipe and Annular Flows

Abstract A theoretical foundation of a new simulator for hydraulic calculations of mud transport is presented. It is demonstrated that it is possible to design a simulator based on theoretical concepts taken from fluid dynamics. The theory is based on a simple turbulence model, which is calibrated against direct simulations of turbulent flows, and a simple expression for suppression of turbulence at low wall strain-rates. The model predicts turbulent non-Newtonian flows, provided rheology data is available. This is an attractive feature of the model as no general non- Newtonian flow correlations are available in the literature. The output from the model is wall shear stresses and pressure drops, while the most important input is the fluid data, and in particular the fluid rheology and its dependency on pressure and temperature. Due to instabilities caused by the drill-string and variable surface roughness at the contacting walls, it is assumed that the flow is unstable or quasi-turbulent, even for low Reynolds numbers well below 1,000. Also, effects of rotation and heat transport are treated in the same fundamental manner. The effects of rotation are included by solving for the tangential velocity in addition to the axial velocity. It is shown that rotation may have a profound impact on the pressure drop. The suggested methodology is expected to increase the precision of predictions and reduce the costs of lab testing and traditional curve fitting to analytical models. The model is verified against experimental data. For most situations, we obtained good agreement with data, as well as for laminar/turbulent transitional flows. Finally, the paper indicates that traditional Fann readings, using 4 points, are inadequate for an accurate rheology determination. In order to improve the hydraulic model, more focus on rheology data is needed. Introduction An important aspect of hydraulic optimization is an accurate estimation of the equivalent circulating density (ECD), especially in long, narrow wells. For this purpose, several simulators for hydraulic calculations of drilling wells have been developed(1). One frequent problem in such simulations is the quality of rheological data that affects the selection of the rheological model. A common approach is to let the user select a rheological model from a set of predefined rheologies(2, 3). Another approach is to curve fit the data to a best fit with a selected rheology model(4). Denis and Guillot(5) found that the Fann viscometer systematically seems to overpredict pressure losses. They tested one pipe viscometer and two rotational viscometers and recommended applying a high quality viscometer instead of the inaccurate Fann viscometer. In another work, Efaghi et al.(6) had access to the training well at Louisiana State University and recorded pressure losses of mud in a 27/8 in., 3,000 ft. deep tubing, comparing Bingham and Power law rheological models. Laminar pressure losses were not predicted accurately since the viscometer shear rate was not representative of the actual laminar flow in the tubing. Valuable data of turbulent flow of non-Newtonian fluids in rough pipes was recorded by Szilas, Bobok, and Navratil(7).

Quasi-Monotone Advection Schemes Based on Explicit Locally Adaptive Dissipation

The authors develop and test computational methods for advection of a scalar field that also include a minimal dissipation of its variance in order to preclude the formation of false extrema. Both of these properties are desirable for advectively dominated geophysical flows, where the relevant scalars are both potential vorticity and material concentrations. These methods are based upon the sequential application of two types of operators: 1) a conservative and nondissipative (i.e., preserving first and second spatial moments of the scalar field), directionally symmetric advection operator with a relatively high order of spatial accuracy; and 2) a locally adaptive correction operator of lower spatial accuracy that eliminates false extrema and causes dissipation. During this correction phase the provisional distribution of the advected quantity is checked against the previous distribution, in order to detect places where the previous values were overshot, and thus to compute the excess. Then an iterative diffusion procedure is applied to the excess field in order to achieve approximate monotone behavior of the solution. In addition to the traditional simple flow tests, we have made long-term simulations of freely evolving two-dimensional turbulent flow in order to compare the performance of the proposed technique with that of previously known algorithms, such as UTOPIA and FCT. This is done for both advection of vorticity and passive scalar. Unlike the simple test flows, the turbulent flow provides nonlinear cascades of quadratic moments of the advected quantities toward small scales, which eventually cannot be resolved on the fixed grid and therefore must be dissipated. Thus, not only the ability of the schemes to produce accurate shape-preserving advection, but also their ability to simulate subgrid-scale dissipation are being compared. It is demonstrated that locally adaptive algorithms designed to avoid oscillatory behavior in the vicinity of steep gradients of the advected scalars may result in overall less dissipation, yet give a locally accurate and physically meaningful solution, whereas algorithms with built-in hyperdiffusion (i.e., those traditionally used for direct simulation of turbulent flows) tend to produce a locally unsufficient and, at the same time, globally excessive amount of dissipation. Finally, the authors assess the practial trade-offs required for large models among the competing attributes of accuracy, extrema preservation, minimal dissipation (e.g., appropriate to large Reynolds numbers), and computational cost.

Development of a two-equation heat transfer model based on direct simulations of turbulent flows with different Prandtl numbers

Low-Reynolds-number type k−ε and kt−εt models have been constructed with the aid of direct numerical simulation (DNS) databases. The proposed models incorporate new velocity and time scales to represent various sizes of eddies in velocity and thermal fields with different Prandtl numbers. The validity of the present k−ε model was tested by application to basic and complex flows such as flows with injection and suction, flows with strong adverse and favorable pressure gradients, and flows with separation and reattachment, while comparing the relevant DNS and reliable experimental data. Fundamental properties of the proposed kt−εt model were first verified in basic flows under arbitrary wall thermal boundary conditions and next in backward-facing step flows at various Prandtl numbers through a comparison of the predictions with the DNS and measurements. These comparisons have proven that the proposed models for both velocity and thermal fields have wide applicability to science and engineering and have sufficient capability to perform highly stable computations at any Prandtl numbers, irrespective of flow configurations.

Development of Two-Equation Heat Transfer Model Based on Direct Simulations of Turbulent Flows with Different Prandtl Numbers.

A low-Reynolds-number two-equation kt-et model has been constructed with the aid of direct numericai simulation (DNS) data bases on turbulent flows with different Prandtl numbers. The model adopts new length and time scales to represent various sizes of eddies in a thermal field. The predicted results for thermal fields at different Prandtl numbers were compared with the relevant DNS and experimental data. These comparisons indicate that the proposed model has sufficient capability to reproduce the DNS data, to perform highly stable computations and to analyze the heat transfer in various Prandtl number fluids.

Direct simulation of turbulent flow in a square duct: Reynolds-stress budgets

The data base from a direct numerical simulation of turbulent flow in a square duct is used to calculate all the terms in the Reynolds stress transport equations. The simulation of this complex turbulent flow was performed at a Reynolds number of 600 based on the friction velocity and the duct width. The distributions of the Reynolds stress budget terms along the wall bisector show similar dynamics to wall-bounded turbulent flows with one inhomogeneous direction. Budget terms in the vicinity of the corner demonstrate how transport and redistribution of energy and shear stresses between the Reynolds stress components takes place, promoting the turbulence characteristics of secondary flows of the second kind. The redistribution of energy by pressure velocity correlations can be explained by the low pressures at the cores of streamwise vortices. The data base is also used to evaluate a nonlinear turbulence model in its ability to accommodate the anisotropy of the Reynolds stress tensor in this flow. This anisotropy is known to be entirely responsible for the formation of the secondary flow in noncircular ducts and cannot be captured by linear models.

Direct simulation of turbulent flow and heat transfer in a channel. Part II: A Green's function technique for wavy walls

AbstractThe possibility of improving heat transfer rates by the imposition of a wavy boundary led to this extension of direct simulation of heat transfer in turbulent flow in a channel. This part of the paper shows how a pattern of micro‐riblets on the surface and parallel to the flow is incorporated into the solution of the Navier‐Stokes and temperature equations. The channel is two‐dimensional with walls at different but uniform temperatures, and the fluid properties are constant. The presence of the patterns presents a considerable computational problem that can be largely overcome by the use of a discrete Green function technique. The method is illustrated by making short‐time calculations of the velocity and temperature fields for a Prandtl number of 0.72, a Reynolds number based on channel half‐width and bulk velocity of 2800, and a sinusoidal pattern with a height of 5 wall units.

Direct simulation of turbulent flow and heat transfer in a channel. Part I: Smooth walls

AbstractInterest in the use of supercomputers for the direct numerical calculation of turbulence prompts the development of efficient numerical techniques so that calculation at higher Reynolds numbers might be made. This paper presents an efficient pseudo‐spectral technique, similar to but different from others that have recently appeared, for the calculation of momentum and heat transfer to a constant‐property, turbulent fluid in a two‐dimensional channel with walls at different, uniform temperature. The code uses no empiricism, although periodic boundary conditions are used for fluctuating quantities in the streamwise and spanwise directions.Calculations were made for a Prandtl number of 0·72 and Reynolds number based on friction velocity and channel half‐height of 180 or 2800 based on channel half‐height and average velocity. Calculations of mean velocity profile, turbulence intensities, skewness, flatness, Reynolds stress and eddy diffusivity of heat near a wall compare favourably with experimental results. Representative contour plots of the temperature field near the wall and of the spanwise and streamwise two‐point velocity correlations are given.Deficiencies are that the calculation requires many hours on a fast computer with a large high‐speed memory and that the grid size in each direction for appropriate resolution is approximately proportional to the square of the Reynolds number and to the Prandtl number raised to some power greater than one.

Direct simulation of fluid flow with cellular automata and the lattice-Boltzmann equation

With the invention of the Hexagonal Lattice Gas it was hoped that this new technique would facilitate direct simulation of turbulent flow. In the past years, however, we have learned about its barriers on numerical accuracy and computational efficiency, which cannot easily be taken. The work on lattice gases has evolved in the introduction of the lattice-Boltzmann scheme. With the appropriate refinements this scheme provides the essential balance between robustness and numerical accuracy and enables us to simulate three-dimensional time-dependent flows at Reynolds numbers up to 50000.

Boundary conditions for direct simulations of compressible viscous flows

Procedures to define boundary conditions for Navier-Stokes equations are discussed. A new formulation using characteristic wave relations through boundaries is derived for the Euler equations and generalized to the Navier-Stokes equations. The emphasis is on deriving boundary conditions compatible with modern non-dissipative algorithms used for direct simulations of turbulent flows. These methods have very low dispersion errors and require precise boundary conditions to avoid numerical instabilities and to control spurious wave reflections at the computational boundaries. The present formulation is an attempt to provide such conditions. Reflecting and non-reflecting boundary condition treatments are presented. Examples of practical implementations for inlet and outlet boundaries as well as slip and no-slip walls are presented. The method applies to subsonic and supersonic flows. It is compared with a reference method based on extrapolation and partial use of Riemann invariants. Test cases described include a ducted shear layer, vortices propagating through boundaries, and Poiseuille flow. Although no mathematical proof of well-posedness is given, the method uses the correct number of boundary conditions required for well-posedness of the Navier-Stokes equations and the examples reveal that it provides a significant improvement over the reference method.

High-order splitting methods for the incompressible Navier-Stokes equations

A new pressure formulation for splitting methods is developed that results in high-order time-accurate schemes for the solution of the incompressible Navier-Stokes equations. In particular, improved pressure boundary conditions of high order in time are introduced that minimize the effect of erroneous numerical boundary layers induced by splitting methods. A new family of stiffly stable schemes is employed in mixed explicit/implicit time-intgration rules. These schemes exhibit much broader stability regions as compared to Adams-family schemes, typically used in splitting methods. Their stability properties remain almost constant as the accuracy of the integration increases, so that robust third- or higher-order time-accurate schemes can readily be constructed that remain stable at relatively large CFL number. The new schemes are implemented within the framework of spectral element discretizations in space so that flexibility and accuracy is guaranteed in the numerical experimentation. A model Stokes problem is studied in detail, and several examples of Navier-Stokes solutions of flows in complex geometries are reported. Comparison is made with the previously used first-order in time spectral element splitting and non-splitting (e.g., Uzawa) schemes. High-order splitting/spectral element methods combine accuracy in space and time, and flexibility in geometry, and thus can be very efficient in direct simulations of turbulent flows in complex geometries.

Direct simulations of turbulent flow using finite-difference schemes

Abstract This paper presents finite-difference solutions to the evolution of small-amplitude disturbances and incompressible fully developed turbulent channel flow. The main objective of the paper is to provide a comprehensive comparison between the results obtained using finite-difference and spectral methods. An advantage of finite-difference schemes over the highly accurate spectral methods lies in the ease with which they can be applied to complex geometries. The finite-difference methods used include a kinetic-energy-conserving type of central difference scheme and a high-order-accurate upwind difference scheme. Unlike the central difference scheme, the upwind difference scheme was found not to require a kinetic energy conservation property to control aliasing error. The dissipative nature of the upwind scheme results in a damping of the higher frequency content. As a result very little energy is aliased back. The computed data (including first- and second-order statistics) for the turbulent channel flow case are found to compare well with experimental data and earlier spectral simulations. It appears that the high-order-accurate upwind scheme is a good candidate for direct simulations of turbulent flows over complex geometries.

Numerical Accuracy of the Direct Simulation of Turbulent Flow.

The accuracy of the direct simulation of turbulent flow is discussed on the basis of the numerical results of two-dimensional homogeneous isotropic turbulent flow. Four cases of calculation have been carried out by pseudospectral methods with/without removal of aliasing error and by finite difference methods with/without consistency between pressure and continuity equation. The grid resolution is required to be finer than 1/4 of the turbulence microscale for the full turbulence simulation. In that case, the finite difference method of more than the 4th order provides a result comparable with that obtained by means of the pseudospectral method. The reduction of the aliasing error is indispensable in the pseudospectral scheme. For the finite difference scheme for the pressure term using the regular mesh, higher accuracy for the Poisson equation is recommended rather than consistency with the continuity equation.

Plane waves and structures in turbulent channel flow

A direct simulation of turbulent flow in a channel is analyzed by the method of empirical eigenfunctions (Karhunen–Loève procedure, proper orthogonal decomposition). This analysis reveals the presence of propagating plane waves in the turbulent flow. The velocity of propagation is determined by the flow velocity at the location of maximal Reynolds stress. The analysis further suggests that the interaction of these waves appears to be essential to the local production of turbulence via bursting or sweeping events in the turbulent boundary layer, with the additional suggestion that the fast acting plane waves act as triggers.

The frequency spectrum of turbulent wall pressure fluctuations: comparison of simulations with experiments

The frequency spectrum of the wall pressure fluctuations obtained from a direct simulation of turbulent flow in a rectangular channel is compared with results from boundary layer experiments. The computed spectrum exhibits ω −1 dependence in a middle frequency range and ω −3·5 dependence at high frequencies. The computed results give good agreement with low Reynolds number experimental results when inner scaling is used.

Renormalization group and local order in strong turbulence

We present an overview of the renormalization group (RNG) method for hydrodynamic turbulence. The method yields both scaling relations and the amplitudes of inertial-range correlation functions. This makes the RNG an effective tool for turbulence modelling and simulations. The physical basis for the RNG analysis is discussed in terms of locally ordered, nearly force-free flow structures recently found in direct simulations of turbulent flows.